Consider an arbitrary \(\tilde{Q} \in \mathcal {Q}(q)\) , and Doob martingale \((Q_t)_{t \ge 0}\)  generated by \(\tilde{Q}\) , i.e., \(Q_t \triangleq \mathbb {E}[ \tilde{Q} | \mathcal {F}_t ]\)  for each \(t\)  (we have \(\lim _{t \rightarrow \infty } Q_t = \tilde{Q}\)  and \(Q_0 = \mathbb {E}\tilde{Q} = q\) ).
By the martingale representation theorem [1]}, there exists a predictable process \(\gamma \)  such that \(Q_t = Q_0 + \int _{s=0}^t \gamma _s dW_s\) .
Since \(Q_t = \mathbb {E}[ \tilde{Q} | \mathcal {F}_t] \in [0,1]\)  for any \(t\) , we have an admissible adversary's policy \(\gamma \in \Gamma (q)\)  and therefore \(\tilde{Q} \in \left\lbrace  Q_\infty ^{q,\gamma } | \gamma \in \Gamma (q) \right\rbrace \)  and \(\mathcal {Q}(q) \subseteq \left\lbrace  Q_\infty ^{q,\gamma } | \gamma \in \Gamma (q) \right\rbrace \) .
